{"pk":64916,"title":"The induced saturation problem for posets","subtitle":null,"abstract":"For a fixed poset \\(P\\), a family \\(\\mathcal F\\) of subsets of \\([n]\\) is induced \\(P\\)-saturated if \\(\\mathcal F\\) does not contain an induced copy of \\(P\\), but for every subset \\(S\\) of \\([n]\\) such that \\( S\\not \\in \\mathcal F\\), \\(P\\) is an induced subposet of \\(\\mathcal F \\cup \\{S\\}\\). The size of the smallest such family \\(\\mathcal F\\) is denoted by \\(\\text{sat}^* (n,P)\\). Keszegh, Lemons, Martin, Pálvölgyi and Patkós [Journal of Combinatorial Theory Series A, 2021] proved that there is a dichotomy of behaviour for this parameter: given any poset \\(P\\), either \\(\\text{sat}^* (n,P)=O(1)\\) or \\(\\text{sat}^* (n,P)\\geq \\log _2 n\\). In this paper we improve this general result showing that either \\(\\text{sat}^* (n,P)=O(1)\\) or \\(\\text{sat}^* (n,P) \\geq \\min\\{ 2 \\sqrt{n}, n/2+1\\}\\). Our proof makes use of a Turán-type result for digraphs.\nCuriously, it remains open as to whether our result is essentially best possible or not. On the one hand, a conjecture of Ivan states that for the so-called diamond poset \\(\\Diamond\\) we have \\(\\text{sat}^* (n,\\Diamond)=\\Theta (\\sqrt{n})\\); so if true this conjecture implies our result is tight up to a multiplicative constant. On the other hand, a conjecture of Keszegh, Lemons, Martin, Pálvölgyi and Patkós states that given any poset \\(P\\), either \\(\\text{sat}^* (n,P)=O(1)\\) or \\(\\text{sat}^* (n,P)\\geq n+1\\). We prove that this latter conjecture is true for a certain class of posets \\(P\\).\n \nMathematics Subject Classifications: 06A07, 05D05\n \nKeywords: Partially ordered sets, saturation, Turán-type problems","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Partially ordered sets"},{"word":"saturation"},{"word":"Turán-type problems"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/3jd9q8x0","frozenauthors":[{"first_name":"Andrea","middle_name":"","last_name":"Freschi","name_suffix":"","institution":"School of Mathematics, University of Birmingham, U.K.","department":""},{"first_name":"Simón","middle_name":"","last_name":"Piga","name_suffix":"","institution":"School of Mathematics, University of Birmingham, U.K.","department":""},{"first_name":"Maryam","middle_name":"","last_name":"Sharifzadeh","name_suffix":"","institution":"Department of Mathematics and Mathematical Statistics, Umeå Universitet, Sweden","department":""},{"first_name":"Andrew","middle_name":"","last_name":"Treglown","name_suffix":"","institution":"School of Mathematics, University of Birmingham, U.K.","department":""}],"date_submitted":"2023-12-22T14:02:27Z","date_accepted":"2023-12-22T14:02:27Z","date_published":"2023-12-22T08:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64916/galley/49726/download/"}]}